Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Linearizing Equations
Wilfrid Laurier University
Terry Sturtevant
Wilfrid Laurier University
December 12, 2014
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Overview
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Overview
In this document, you’ll learn:
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Overview
In this document, you’ll learn:
what it means to linearize an equation
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Overview
In this document, you’ll learn:
what it means to linearize an equation
when to do it
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Overview
In this document, you’ll learn:
what it means to linearize an equation
when to do it
why it’s useful
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Overview
In this document, you’ll learn:
what it means to linearize an equation
when to do it
why it’s useful
how to handle uncertainties when linearizing
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Suppose we have a marble, and we want to calculate its density:
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Suppose we have a marble, and we want to calculate its density:
m
mass m = 5 g
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Suppose we have a marble, and we want to calculate its density:
m
d = 2r
mass m = 5 g
radius r = 0.7 cm; (diameter is easier to measure than radius)
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Suppose we have a marble, and we want to calculate its density:
m
d = 2r
mass m = 5 g
radius r = 0.7 cm
density of the marble ρ = m/v and v = 43 πr 3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Suppose we have a marble, and we want to calculate its density:
m
d = 2r
mass m = 5 g
radius r = 0.7 cm
density of the marble ρ = m/v and v = 43 πr 3
Thus ρ =
=
m
4
πr 3
3
3×0.05
4π(0.7×10−2 )3
=
3m
4πr 3
(in Si units)
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Since radius changes, we could take several values of r :
m
mass m = 5 g
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Since radius changes, we could take several values of r :
m
d1 mass m = 5 g
radius r1 = 0.7cm
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Since radius changes, we could take several values of r :
m
d
2
d1 mass m = 5 g
radius r1 = 0.7cm, r2 = 0.68cm
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Since radius changes, we could take several values of r :
m
d
2
d1 d3
mass m = 5 g
radius r1 = 0.7cm, r2 = 0.68cm, and r3 = 0.71cm.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Since radius changes, we could take several values of r :
m
d
2
d1 d3
mass m = 5 g
radius r1 = 0.7cm, r2 = 0.68cm, and r3 = 0.71cm.
Average the r values, and proceed as before.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Since radius changes, we could take several values of r :
m
d
2
d1 d3
mass m = 5 g
radius r1 = 0.7cm, r2 = 0.68cm, and r3 = 0.71cm.
Average the r values, and proceed as before.
r̄ =
r1 +r2 +r3
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Since radius changes, we could take several values of r :
m
d
2
d1 d3
mass m = 5 g
radius r1 = 0.7cm, r2 = 0.68cm, and r3 = 0.71cm.
Average the r values, and proceed as before.
r1 +r2 +r3
3
3m
Thus ρ = 4πr̄
3
r̄ =
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
What if we had several marbles of the same material?
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
What if we had several marbles of the same material?
m1 , r1
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
What if we had several marbles of the same material?
m 2 , r2
m1 , r1
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
What if we had several marbles of the same material?
m 2 , r2
m1 , r1
m3 , r3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
What if we had several marbles of the same material?
m 2 , r2
m1 , r1
m3 , r3
We could average the m values and average the r values so
that
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
What if we had several marbles of the same material?
m 2 , r2
m1 , r1
m3 , r3
We could average the m values and average the r values so
that
ρ=
3m̄
4πr̄ 3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
What if we had several marbles of the same material?
m 2 , r2
m1 , r1
m3 , r3
We could average the m values and average the r values so
that
ρ=
3m̄
4πr̄ 3
This is not the best solution. (The small marble will have
almost no effect on the calculation.)
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Q:Why not just calculate the density for each of the marbles
and average the results?
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Q:Why not just calculate the density for each of the marbles
and average the results?
A:The proportional uncertainty in the mass and radius of the
small marble will be much bigger than the others, and so it
could have too large an influence on the result.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Q:Why not just calculate the density for each of the marbles
and average the results?
A:The proportional uncertainty in the mass and radius of the
small marble will be much bigger than the others, and so it
could have too large an influence on the result.
Is there some way to prevent one data point from having
a disproportionate effect on the result?
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Any method of calculating the density, ρ, is based on the
mathematical relationship between m and r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Any method of calculating the density, ρ, is based on the
mathematical relationship between m and r
We can show a mathematical relationship between variables
by a graph.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
m
r
The graph of m vs. r is not linear.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
m ∝ r3
m
r
The graph of m vs. r is not linear.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
If we have a straight line graph, then knowing the slope and
the y -intercept tells us everything we need to know about the
relationship between the variables.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
If we have a straight line graph, then knowing the slope and
the y -intercept tells us everything we need to know about the
relationship between the variables.
i.e. Y = MX + B
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
If we have a straight line graph, then knowing the slope and
the y -intercept tells us everything we need to know about the
relationship between the variables.
i.e. Y = MX + B
So we have var = constant
Terry Sturtevant
var + constant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
If we have a straight line graph, then knowing the slope and
the y -intercept tells us everything we need to know about the
relationship between the variables.
i.e. Y = MX + B
So we have var = constant
var + constant
Knowing the two constants, M and B, tells us how to
calculate one variable given the other.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Linearizing equations is the process of modifying an equation to
produce new variables which can be plotted to produce a straight
line graph.
Whenever we can turn an equation into the form
var = constant
var + constant
we have linearized it.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Linearizing equations is the process of modifying an equation to
produce new variables which can be plotted to produce a straight
line graph.
Whenever we can turn an equation into the form
var = constant
var + constant
we have linearized it.
A plot is better than an average since it may indicate
systematic errors in the data.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Linearizing equations is the process of modifying an equation to
produce new variables which can be plotted to produce a straight
line graph.
Whenever we can turn an equation into the form
var = constant
var + constant
we have linearized it.
A plot is better than an average since it may indicate
systematic errors in the data.
A straight line is easy to spot with the unaided eye.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Linearizing equations is the process of modifying an equation to
produce new variables which can be plotted to produce a straight
line graph.
Whenever we can turn an equation into the form
var = constant
var + constant
we have linearized it.
A plot is better than an average since it may indicate
systematic errors in the data.
A straight line is easy to spot with the unaided eye.
A fit equation replaces a bunch of data with a few
parameters: M, ∆M, B and ∆B.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Combining constants gives a constant.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Combining constants gives a constant.
e.g. 2π is a constant.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Combining constants gives a constant.
e.g. 2π is a constant.
Functions of variables are variables.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Combining constants gives a constant.
e.g. 2π is a constant.
Functions of variables are variables.
√
e.g. r is a variable if r is a variable.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Combining constants gives a constant.
e.g. 2π is a constant.
Functions of variables are variables.
√
e.g. r is a variable if r is a variable.
Combining variables gives a variable.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Combining constants gives a constant.
e.g. 2π is a constant.
Functions of variables are variables.
√
e.g. r is a variable if r is a variable.
Combining variables gives a variable.
e.g. v =
x
t
is a variable if x and t are variables.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Combining constants gives a constant.
e.g. 2π is a constant.
Functions of variables are variables.
√
e.g. r is a variable if r is a variable.
Combining variables gives a variable.
e.g. v =
x
t
is a variable if x and t are variables.
Combining constants and variables gives a variable.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Rules about constants and variables: (A constant is the same for
all data values; a variable changes for different data values.)
Functions of constants are constants.
√
e.g. π is a constant
Combining constants gives a constant.
e.g. 2π is a constant.
Functions of variables are variables.
√
e.g. r is a variable if r is a variable.
Combining variables gives a variable.
e.g. v =
x
t
is a variable if x and t are variables.
Combining constants and variables gives a variable.
e.g. d = 2r is a variable if r is a variable.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Terry Sturtevant
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
4πρr 3
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
or m
4πρr 3
3
3
= 4πρ
3 r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
or m
4πρr 3
3
3
= 4πρ
3 r
A better way to see this is
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
or m
4πρr 3
3
3
= 4πρ
3 r
A better way to see this is
m=
4πρ 3
3 r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
or m
4πρr 3
3
3
= 4πρ
3 r
A better way to see this is
m=
4πρ 3
3 r
Note that
4πρ
3
is a constant.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
or m
4πρr 3
3
3
= 4πρ
3 r
A better way to see this is
m=
4πρ 3
3 r
Note that
4πρ
3
is a constant.
So we have var = constant
Terry Sturtevant
var
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
or m
4πρr 3
3
3
= 4πρ
3 r
A better way to see this is
m=
4πρ 3
3 r
Note that
4πρ
3
is a constant.
So we have var = constant
or var = constant
var
var + 0
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
or m
4πρr 3
3
3
= 4πρ
3 r
A better way to see this is
m=
4πρ 3
3 r
Note that
4πρ
3
is a constant.
So we have var = constant
or var = constant
var
var + 0
which is the equation of a straight line
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Density of the marble ρ =
Calculations with multiple data points
Mathematical relationships between variables
3m
4πr 3
This could be rearranged to solve for m, so
m=
or m
4πρr 3
3
3
= 4πρ
3 r
A better way to see this is
m=
4πρ 3
3 r
Note that
4πρ
3
is a constant.
So we have var = constant
or var = constant
var
var + 0
which is the equation of a straight line
i.e. Y = MX + B
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
m=
4πρ 3
3 r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
m=
4πρ 3
3 r
is the equation of a straight line
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
m=
4πρ 3
3 r
is the equation of a straight line
i.e. Y = MX + B
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
m=
4πρ 3
3 r
is the equation of a straight line
i.e. Y = MX + B
The slope, M =
4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
m=
4πρ 3
3 r
is the equation of a straight line
i.e. Y = MX + B
The slope, M =
4πρ
3
The y -intercept, B, should be zero.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
m=
4πρ 3
3 r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
m=
4πρ 3
3 r
Plot m versus r 3 .
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
m=
4πρ 3
3 r
Plot m versus r 3 .
The slope, M =
4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
m=
4πρ 3
3 r
Plot m versus r 3 .
The slope, M =
4πρ
3
Rearranging this gives
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
m=
4πρ 3
3 r
Plot m versus r 3 .
The slope, M =
4πρ
3
Rearranging this gives
ρ=
3M
4π
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
m=
4πρ 3
3 r
Plot m versus r 3 .
The slope, M =
4πρ
3
Rearranging this gives
ρ=
3M
4π
So the density of the marble ρ =
Terry Sturtevant
3×(slope)
4π
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
m=
4πρ 3
3 r
Plot m versus r 3 .
The slope, M =
4πρ
3
Rearranging this gives
ρ=
3M
4π
So the density of the marble ρ =
3×(slope)
4π
Note that after linearization, the original variables, m
and r , are gone. The density only depends on the slope.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
+ ln r 3
ln (m) = ln 4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
+ ln r 3
ln (m) = ln 4πρ
3
ln (m) = ln 4πρ
+ 3 ln (r )
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
+ ln r 3
ln (m) = ln 4πρ
3
ln (m) = ln 4πρ
+ 3 ln (r )
3
We could show it this way instead:
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
+ ln r 3
ln (m) = ln 4πρ
3
ln (m) = ln 4πρ
+ 3 ln (r )
3
We could show it this way instead:
ln (m) = 3ln (r ) + ln 4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
+ ln r 3
ln (m) = ln 4πρ
3
ln (m) = ln 4πρ
+ 3 ln (r )
3
We could show it this way instead:
ln (m) = 3ln (r ) + ln 4πρ
3
Note that 3 and
4πρ
3
are constants.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
+ ln r 3
ln (m) = ln 4πρ
3
ln (m) = ln 4πρ
+ 3 ln (r )
3
We could show it this way instead:
ln (m) = 3ln (r ) + ln 4πρ
3
Note that 3 and 4πρ
3 are constants.
So we have var = constant var + constant
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
+ ln r 3
ln (m) = ln 4πρ
3
ln (m) = ln 4πρ
+ 3 ln (r )
3
We could show it this way instead:
ln (m) = 3ln (r ) + ln 4πρ
3
Note that 3 and 4πρ
3 are constants.
So we have var = constant var + constant
which is the equation of a straight line
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s another possible linearization:
3
m = 4πρ
3 r
We could take logarithms of both sides to give
3
ln (m) = ln 4πρ
3 r
+ ln r 3
ln (m) = ln 4πρ
3
ln (m) = ln 4πρ
+ 3 ln (r )
3
We could show it this way instead:
ln (m) = 3ln (r ) + ln 4πρ
3
Note that 3 and 4πρ
3 are constants.
So we have var = constant var + constant
which is the equation of a straight line
i.e. Y = MX + B
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
ln (m) = 3ln (r ) + ln
4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
ln (m) = 3ln (r ) + ln
4πρ
3
is the equation of a straight line
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
ln (m) = 3ln (r ) + ln
4πρ
3
is the equation of a straight line
i.e. Y = MX + B
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
ln (m) = 3ln (r ) + ln
4πρ
3
is the equation of a straight line
i.e. Y = MX + B
The slope, M = 3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
To summarize:
ln (m) = 3ln (r ) + ln
4πρ
3
is the equation of a straight line
i.e. Y = MX + B
The slope, M = 3
The y -intercept, B = ln
4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
ln (m) = 3ln (r ) + ln 4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
ln (m) = 3ln (r ) + ln 4πρ
3
Plot ln m versus ln r .
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
ln (m) = 3ln (r ) + ln 4πρ
3
Plot ln m versus ln r .
The y -intercept, B = ln
4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
ln (m) = 3ln (r ) + ln 4πρ
3
Plot ln m versus ln r .
The y -intercept, B = ln
4πρ
3
Rearranging this gives
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
ln (m) = 3ln (r ) + ln 4πρ
3
Plot ln m versus ln r .
The y -intercept, B = ln
4πρ
3
Rearranging this gives
e B = 4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
ln (m) = 3ln (r ) + ln 4πρ
3
Plot ln m versus ln r .
The y -intercept, B = ln
4πρ
3
Rearranging this gives
e B = 4πρ
3
and so ρ =
3e B
4π
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
ln (m) = 3ln (r ) + ln 4πρ
3
Plot ln m versus ln r .
The y -intercept, B = ln
4πρ
3
Rearranging this gives
e B = 4πρ
3
and so ρ =
3e B
4π
So the density of the marble ρ =
Terry Sturtevant
3e y −intercept
4π
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
So how do we get the density from the graph?
ln (m) = 3ln (r ) + ln 4πρ
3
Plot ln m versus ln r .
The y -intercept, B = ln
4πρ
3
Rearranging this gives
e B = 4πρ
3
and so ρ =
3e B
4π
So the density of the marble ρ =
3e y −intercept
4π
Note that after linearization, the original variables, m
and r , are gone. The density only depends on the
y -intercept.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
Here’s how the graphs look for both linearizations.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
m
r3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
m=
Calculations with multiple data points
Mathematical relationships between variables
4πρ 3
3 r
m
r3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
m=
Calculations with multiple data points
Mathematical relationships between variables
4πρ 3
3 r
m
@
I M = 4πρ
3
r3
The density can be calculated from the slope.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
m=
Calculations with multiple data points
Mathematical relationships between variables
4πρ 3
3 r
m
@
I M = 4πρ
3
B=0
r3
The density can be calculated from the slope. The y -intercept isn’t
zero, so there may be a systematic error.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
ln (m)
ln (r )
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
ln (m) = 3ln (r ) + ln
4πρ
3
#
#
#
#
#
#
#
#
ln (m)
#
#
#
#
#
#
#
ln (r )
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
ln (m) = 3ln (r ) + ln
4πρ
3
#
#
#
#
#
#
#
#
ln (m)
#
#
#
#
#
4πρ
B
=
ln
#
3
# 9
ln (r )
The density can be calculated from the y -intercept.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Calculations with multiple data points
Mathematical relationships between variables
ln (m) = 3ln (r ) + ln
4πρ
3
#
#
#
#
#
#
#
#
ln (m)
#
#
#
#
I M=3
@
#
4πρ
#
B = ln 3
# 9
ln (r )
The density can be calculated from the y -intercept. The slope
should be 3, so this can be checked to look for systematic error.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
All of the calculations follow the same rules as for any other
calculations. There are two specific types of calculations:
1
Uncertainties in linearized variables
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
All of the calculations follow the same rules as for any other
calculations. There are two specific types of calculations:
1
Uncertainties in linearized variables
2
Uncertainties in results of the linearization
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
All of the calculations follow the same rules as for any other
calculations. There are two specific types of calculations:
1
Uncertainties in linearized variables
2
Uncertainties in results of the linearization
We’ll look at the earlier examples
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
m=
4πρ 3
3 r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
4πρ 3
3 r
plot r 3 as
m=
To
a variable, we need to determine ∆ r 3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
4πρ 3
3 r
plot r 3 as
m=
a variable, we need to determine ∆ r 3
3
2
By the rule for functions, ∆ r 3 ≈ dr
dr ∆r = 3r ∆r
To
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
4πρ 3
3 r
plot r 3 as
m=
a variable, we need to determine ∆ r 3
3
2
By the rule for functions, ∆ r 3 ≈ dr
dr ∆r = 3r ∆r
To
The slope, M =
4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
4πρ 3
3 r
plot r 3 as
m=
a variable, we need to determine ∆ r 3
3
2
By the rule for functions, ∆ r 3 ≈ dr
dr ∆r = 3r ∆r
To
The slope, M =
4πρ
3
Rearranging this gives
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
4πρ 3
3 r
plot r 3 as
m=
a variable, we need to determine ∆ r 3
3
2
By the rule for functions, ∆ r 3 ≈ dr
dr ∆r = 3r ∆r
To
The slope, M =
4πρ
3
Rearranging this gives
3
ρ = 3M
4π = 4π M
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
4πρ 3
3 r
plot r 3 as
m=
a variable, we need to determine ∆ r 3
3
2
By the rule for functions, ∆ r 3 ≈ dr
dr ∆r = 3r ∆r
To
The slope, M =
4πρ
3
Rearranging this gives
3
ρ = 3M
4π = 4π M
3
∆M
and so ∆ρ = 4π
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
In the first case we have
4πρ 3
3 r
plot r 3 as
m=
a variable, we need to determine ∆ r 3
3
2
By the rule for functions, ∆ r 3 ≈ dr
dr ∆r = 3r ∆r
To
The slope, M =
4πρ
3
Rearranging this gives
3
ρ = 3M
4π = 4π M
3
∆M
and so ∆ρ = 4π
As before, after linearization, the original uncertainties,
∆m and ∆r , are gone. The uncertainty in the density
only depends on the uncertainty in the slope.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
ln (m) = 3ln (r ) + ln 4πρ
3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
ln (m) = 3ln (r ) + ln 4πρ
3
To plot ln (m) as a variable, we need to determine ∆ ln (m)
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
ln (m) = 3ln (r ) + ln 4πρ
3
To plot ln (m) as a variable, we need to determine ∆ ln (m)
and to plot ln (r ) as a variable, we need to determine ∆ ln (r )
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
ln (m) = 3ln (r ) + ln 4πρ
3
To plot ln (m) as a variable, we need to determine ∆ ln (m)
and to plot ln (r ) as a variable, we need to determine ∆ ln (r )
By the rule for functions,
(m)
∆ ln (m) ≈ dlndm
∆m = m1 ∆m =
Terry Sturtevant
∆m
m
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
ln (m) = 3ln (r ) + ln 4πρ
3
To plot ln (m) as a variable, we need to determine ∆ ln (m)
and to plot ln (r ) as a variable, we need to determine ∆ ln (r )
By the rule for functions,
(m)
∆ ln (m) ≈ dlndm
∆m = m1 ∆m =
and similarly ∆ ln (r ) ≈
∆m
m
∆r
r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
ln (m) = 3ln (r ) + ln 4πρ
3
To plot ln (m) as a variable, we need to determine ∆ ln (m)
and to plot ln (r ) as a variable, we need to determine ∆ ln (r )
By the rule for functions,
(m)
∆ ln (m) ≈ dlndm
∆m = m1 ∆m =
and similarly ∆ ln (r ) ≈
∆m
m
∆r
r
The density of the marble ρ =
Terry Sturtevant
3e B
4π
=
3 B
4π e
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
ln (m) = 3ln (r ) + ln 4πρ
3
To plot ln (m) as a variable, we need to determine ∆ ln (m)
and to plot ln (r ) as a variable, we need to determine ∆ ln (r )
By the rule for functions,
(m)
∆ ln (m) ≈ dlndm
∆m = m1 ∆m =
and similarly ∆ ln (r ) ≈
∆r
r
The density of the marble ρ =
so ∆ρ ≈
3 B
d 4π
e
dB ∆B
=
∆m
m
3e B
4π
3 de B
4π dB ∆B
Terry Sturtevant
=
=
3 B
4π e
3 B
4π e ∆B
= ρ∆B!
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For the other linearization:
ln (m) = 3ln (r ) + ln 4πρ
3
To plot ln (m) as a variable, we need to determine ∆ ln (m)
and to plot ln (r ) as a variable, we need to determine ∆ ln (r )
By the rule for functions,
(m)
∆ ln (m) ≈ dlndm
∆m = m1 ∆m =
and similarly ∆ ln (r ) ≈
∆r
r
The density of the marble ρ =
so ∆ρ ≈
3 B
d 4π
e
dB ∆B
=
∆m
m
3e B
4π
3 de B
4π dB ∆B
=
=
3 B
4π e
3 B
4π e ∆B
= ρ∆B!
As before, after linearization, the original uncertainties,
∆m and ∆r , are gone. The uncertainty in the density
only depends on the uncertainty in the y -intercept.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For different linearizations, because the variables are different,
the uncertainties are different.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For different linearizations, because the variables are different,
the uncertainties are different.
Take a look at the two sample linearizations.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For different linearizations, because the variables are different,
the uncertainties are different.
Take a look at the two sample linearizations.
The explanations are given for r as an example.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
For different linearizations, because the variables are different,
the uncertainties are different.
Take a look at the two sample linearizations.
The explanations are given for r as an example.
(If ∆r is the same for all of the values it’s easiest to see the
result.)
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Terry Sturtevant
Uncertainty calculations
Error bars
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
m=
Uncertainty calculations
Error bars
4πρ 3
3 r
m
r3
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
m=
Uncertainty calculations
Error bars
4πρ 3
3 r
m
6
r3
∆m
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
m=
Uncertainty calculations
Error bars
4πρ 3
3 r
m
6
2
3r ∆r
r3
∆m
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
m=
Uncertainty calculations
Error bars
4πρ 3
3 r
m
6
2
3r ∆r
r3
∆m
Error bars get bigger as r gets bigger, since ∆r 3 ≈ 3r 2 ∆r
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Terry Sturtevant
Uncertainty calculations
Error bars
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
ln (m)
ln (r )
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
ln (m) = 3ln (r ) + ln
4πρ
3
ln (m)
ln (r )
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
ln (m) = 3ln (r ) + ln
4πρ
3
#
#
#
#
#
#
#
#
ln (m)
#
#
#
#
#
#
#
ln (r )
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
ln (m) = 3ln (r ) + ln
4πρ
3
#
#
#
#
#
#
#
#
ln (m)
#
∆m 6#
m #
#
#
#
#
ln (r )
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
ln (m) = 3ln (r ) + ln
4πρ
3
#
#
#
#
#
#
#
#
ln (m)
#
#
#
∆m 6#
m # # ∆r
#
r
ln (r )
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Uncertainty calculations
Error bars
ln (m) = 3ln (r ) + ln
4πρ
3
#
#
#
#
#
#
#
#
ln (m)
#
#
#
∆m 6#
m # # ∆r
#
r
ln (r )
Error bars get smaller as m and r get bigger, since ∆ ln (r ) ≈
and ∆ ln (m) ≈ ∆m
m
Terry Sturtevant
∆r
r
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Recap
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Recap
1
Linearizing an equation means rearranging it so that you can
make it into a straight line graph.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Recap
1
Linearizing an equation means rearranging it so that you can
make it into a straight line graph.
2
A linearized graph is convenient to use, and makes it easier to
spot possible errors.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Recap
1
Linearizing an equation means rearranging it so that you can
make it into a straight line graph.
2
A linearized graph is convenient to use, and makes it easier to
spot possible errors.
3
After linearization, the only parameters left are the slope,
y -intercept, and their uncertainties.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University
Overview
Linearizing equations
Uncertainties and linearized equations
Recap
Recap
1
Linearizing an equation means rearranging it so that you can
make it into a straight line graph.
2
A linearized graph is convenient to use, and makes it easier to
spot possible errors.
3
After linearization, the only parameters left are the slope,
y -intercept, and their uncertainties.
4
There are often several different linearizations for a single
equation.
Terry Sturtevant
Linearizing Equations Wilfrid Laurier University